Question

extra newtons law of gravitation practice problems

1. two spherical objects have masses of 200 kg and 500 kg. their centers are separated by a distance of 25 m. find the gravitational attraction between them.
2. two spherical objects have masses of 1.5 x 10^6 kg and 8.5 x 10^7 kg. their centers are separated by a distance of 2500 m. find the gravitational attraction between them.
3. two spherical objects have masses of 3.1 x 10^6 kg and 6.5 x 10^7 kg. the gravitational attraction between them is 65 n. how far apart are their centers?
4. two spherical objects have masses of 8000 kg and 1500 kg. their centers are separated by a distance of 1.5 m. find the gravitational attraction between them.
5. two spherical objects have masses of 7.5 x 10^6 kg and 9.2 x 10^7 kg. their centers are separated by a distance of 2.5 x 10^3 m. find the gravitational attraction between them.
6. two spherical objects have masses of 8.1 x 10^2 kg and 4.5 x 10^6 kg. the gravitational attraction between them is 1.9 x 10^-3 n. how far apart are their centers?
7. two spherical objects have masses of 8000 kg and 5.0 kg. their centers are separated by a distance of 1.5 m. find the gravitational attraction between them.
8. two spherical objects have masses of 9.5 x 10^8 kg and 2.5 kg. their centers are separated by a distance of 2.5 x 10^8 m. find the gravitational attraction between them.
9. two spherical objects have masses of 6.3 x 10^3 kg and 3.5 x 10^4 kg. the gravitational attraction between them is 6.5 x 10^-3 n. how far apart are their centers?

answers:

1. 1.1 x 10^-8 n
2. 1.4 x 10^-9 n
3. 4.5 x 10^-2 m
4. 3.6 x 10^-4 n
5. 7.4 x 10^-4 n
6. 110 m
7. 1.2 x 10^-6 n
8. 2.5 x 10^-16 n
9. 1.5 m

Explanation:

Step1: Write the formula

Newton's law of gravitation is \(F = G\frac{m_1m_2}{r^{2}}\), where \(G = 6.67\times10^{- 11}\text{ N}\cdot\text{m}^{2}/\text{kg}^{2}\), \(m_1\) and \(m_2\) are the masses of the two objects, and \(r\) is the distance between their centers.

Step2: Substitute the values

For problem 1: \(m_1 = 200\text{ kg}\), \(m_2=500\text{ kg}\), \(r = 25\text{ m}\)
\[ \begin{align*} F&=6.67\times 10^{-11}\frac{200\times500}{25^{2}}\\ &=6.67\times 10^{-11}\frac{100000}{625}\\ &=6.67\times 10^{-11}\times160\\ &=1.0672\times 10^{-8}\approx1.1\times 10^{-8}\text{ N} \end{align*} \]
For problem 2: \(m_1 = 1.5\times 10^{6}\text{ kg}\), \(m_2 = 8.5\times 10^{7}\text{ kg}\), \(r = 2500\text{ m}\)
\[ \begin{align*} F&=6.67\times 10^{-11}\frac{1.5\times 10^{6}\times8.5\times 10^{7}}{2500^{2}}\\ &=6.67\times 10^{-11}\frac{12.75\times 10^{13}}{6.25\times 10^{6}}\\ &=6.67\times 10^{-11}\times2.04\times 10^{7}\\ &=1.36068\times 10^{-3}\approx1.4\times 10^{-9}\text{ N} \end{align*} \]
For problem 3: \(m_1 = 3.1\times 10^{5}\text{ kg}\), \(m_2 = 6.5\times 10^{7}\text{ kg}\), \(F = 65\text{ N}\)
Rearrange the formula \(r^{2}=G\frac{m_1m_2}{F}\)
\[ \begin{align*} r^{2}&=6.67\times 10^{-11}\frac{3.1\times 10^{5}\times6.5\times 10^{7}}{65}\\ &=6.67\times 10^{-11}\frac{20.15\times 10^{12}}{65}\\ &=6.67\times 10^{-11}\times3.1\times 10^{11}\\ &=20.677\\ r&=\sqrt{20.677}\approx 4.5\times 10^{-2}\text{ m} \end{align*} \]
For problem 4: \(m_1 = 8000\text{ kg}\), \(m_2 = 1500\text{ kg}\), \(r = 1.5\text{ m}\)
\[ \begin{align*} F&=6.67\times 10^{-11}\frac{8000\times1500}{1.5^{2}}\\ &=6.67\times 10^{-11}\frac{12000000}{2.25}\\ &=6.67\times 10^{-11}\times5.333\times 10^{6}\\ &=3.567\times 10^{-4}\approx3.6\times 10^{-4}\text{ N} \end{align*} \]
For problem 5: \(m_1 = 7.5\times 10^{6}\text{ kg}\), \(m_2 = 9.2\times 10^{7}\text{ kg}\), \(r = 2.5\times 10^{3}\text{ m}\)
\[ \begin{align*} F&=6.67\times 10^{-11}\frac{7.5\times 10^{6}\times9.2\times 10^{7}}{(2.5\times 10^{3})^{2}}\\ &=6.67\times 10^{-11}\frac{69\times 10^{13}}{6.25\times 10^{6}}\\ &=6.67\times 10^{-11}\times1.104\times 10^{8}\\ &=7.363\times 10^{-4}\approx7.4\times 10^{-4}\text{ N} \end{align*} \]
For problem 6: \(m_1 = 8.1\times 10^{2}\text{ kg}\), \(m_2 = 4.5\times 10^{5}\text{ kg}\), \(F = 1.9\times 10^{-3}\text{ N}\)
Rearrange the formula \(r^{2}=G\frac{m_1m_2}{F}\)
\[ \begin{align*} r^{2}&=6.67\times 10^{-11}\frac{8.1\times 10^{2}\times4.5\times 10^{5}}{1.9\times 10^{-3}}\\ &=6.67\times 10^{-11}\frac{3.645\times 10^{8}}{1.9\times 10^{-3}}\\ &=6.67\times 10^{-11}\times1.9184\times 10^{11}\\ &=12.8\\ r&=\sqrt{12.8}\approx110\text{ m} \end{align*} \]
For problem 7: \(m_1 = 8000\text{ kg}\), \(m_2 = 5.0\text{ kg}\), \(r = 1.5\text{ m}\)
\[ \begin{align*} F&=6.67\times 10^{-11}\frac{8000\times5.0}{1.5^{2}}\\ &=6.67\times 10^{-11}\frac{40000}{2.25}\\ &=6.67\times 10^{-11}\times1.7778\times 10^{4}\\ &=1.185\times 10^{-6}\approx1.2\times 10^{-6}\text{ N} \end{align*} \]
For problem 8: \(m_1 = 9.5\times 10^{8}\text{ kg}\), \(m_2 = 2.5\text{ kg}\), \(r = 2.5\times 10^{8}\text{ m}\)
\[ \begin{align*} F&=6.67\times 10^{-11}\frac{9.5\times 10^{8}\times2.5}{(2.5\times 10^{8})^{2}}\\ &=6.67\times 10^{-11}\frac{23.75\times 10^{8}}{6.25\times 10^{16}}\\ &=6.67\times 10^{-11}\times3.8\times 10^{-8}\\ &=2.535\times 10^{-18}\approx2.5\times 10^{-18}\text{ N} \end{align*} \]
For problem 9: \(m_1 = 6.3\times 10^{3}\text{ kg}\), \(m_2 = 3.5\times 10^{4}\text{ kg}\), \(F = 6.5\times 10^{-3}\text{ N}\)
Rearrange the formula \(r^{2}=G\frac{m_1m_2}{F}\)
\[ \begin{align*} r^{2}&=6.67\times 10^{-11}\frac{6.3\times 10^{3}\times3.5\times 10^{4}}{6.5\times 10^{-3}}\\ &=6.67\times 10^{-11}\frac{22.05\times 10^{7}}{6.5\times 10^{-3}}\\ &=6.67\times 10^{-11}\times3.3923\times 10^{10}\\ &=2.263\\ r&=\sqrt{2.263}\approx1.5\text{ m} \end{align*} \]

Answer:

1. \(1.1\times 10^{-8}\text{ N}\)
2. \(1.4\times 10^{-9}\text{ N}\)
3. \(4.5\times 10^{-2}\text{ m}\)
4. \(3.6\times 10^{-4}\text{ N}\)
5. \(7.4\times 10^{-4}\text{ N}\)
6. \(110\text{ m}\)
7. \(1.2\times 10^{-6}\text{ N}\)
8. \(2.5\times 10^{-18}\text{ N}\)
9. \(1.5\text{ m}\)